<h2>Problem 184</h2>
<div style="color:#666;font-size:80%;">29 February 2008</div><br />
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<p>Consider the set <var>I<img src="" style="display:none;" alt="_(" /><sub>r</sub><img src="" style="display:none;" alt=")" /></var> of points (<var>x</var>,<var>y</var>) with integer co-ordinates in the interior of the circle with radius <var>r</var>, centered at the origin, i.e. <var>x</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> + <var>y</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> <var>r</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />.</p>
<p>For a radius of 2, <var>I</var><img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> contains the nine points (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1), (0,-1) and (1,-1). There are eight triangles having all three vertices in <var>I</var><img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> which contain the origin in the interior. Two of them are shown below, the others are obtained from these by rotation.</p>
<p style="margin-left:240px;"><img src="project/images/p_184.gif" alt="" /></p>

<p>For a radius of 3, there are 360 triangles containing the origin in the interior and having all vertices in <var>I</var><img src="" style="display:none;" alt="_(" /><sub>3</sub><img src="" style="display:none;" alt=")" /> and for <var>I</var><img src="" style="display:none;" alt="_(" /><sub>5</sub><img src="" style="display:none;" alt=")" /> the number is 10600.</p>

<p>How many triangles are there containing the origin in the interior and having all three vertices in <var>I</var><img src="" style="display:none;" alt="_(" /><sub>105</sub><img src="" style="display:none;" alt=")" />?</p>

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